 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/39b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

u = A e^{i (K \cdot x - ω t)}

K \times (K \times A) = K (K \cdot A) - K^{2} A

gives

ρ ω^{2} A = (λ + μ) K (K \cdot A) + μ K^{2} A .

A

is parallel to

K

, this becomes

ω^{2} = c_{P}^{2} K^{2}, c_{P} = \frac{λ + 2 μ}{ρ},

and gives a longitudinal P-wave. If

A \cdot K = 0

, it becomes

ω^{2} = c_{S}^{2} K^{2}, c_{S} = \frac{μ}{ρ},

and gives either polarization of an S-wave.

For

K = (k, 0, m)

wave P SV SH A (k, 0, m) (m, 0, - k) (0, 1, 0) .

The SV-wave lies in the

x z

propagation plane, whereas the SH-wave is perpendicular to that plane.

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